While all the answers so far are correct, the OP's question seems to have two spaces, thus implying the need to use a two-word tense construction.

There are two compound past tense forms that are acceptable in this context. The more fitting one is an alternate form of the past simple:

No, I haven't, but my brother did lose his camera on a trip once.

This form uses a participle of the actual verb (almost always formed by taking the infinitive minus the word "to"), plus the past simple tense of "do" (which is always "did" regardless of subject) to form the past simple tense. In the general case, this form is synonymous with the conjugated past simple ("lost"), but it typically has the connotation of either contradicting a prior negative (in this case "no I haven't"), and/or emphasizing that the event indeed happened. I can find no online reference defining this construction as anything other than past simple tense, and oddly I can't find any reference that uses the "do" verb to construct this tense in the affirmative (only the negative), but it definitely exists in modern speech.

The other possibility, much less of a fit, is the past perfect or pluperfect:

No, I haven't, but my brother has lost his camera on a trip once.

This is marginal; while it would work to express the fact that your brother, at one point in the indefinite past, lost a camera, the word "once" feels redundant; the use of the past perfect coupled with "once" makes it seem like the count of the number of times it's happened (once) is more important to the speaker than the event (losing a camera). If the sentence instead ended with "many times", this pluperfect tense would be the only valid answer, but as it is it feels a little clunky.

While all the answers so far are correct, the OP's question seems to have two spaces, thus implying the need to use a two-word tense construction.

There are two compound past tense forms that are acceptable in this context. The more fitting one is an alternate form of the past simple:

No, I haven't, but my brother did lose his camera on a trip once.

This form uses a participle of the actual verb (almost always formed by taking the infinitive minus the word "to"), plus the past simple tense of "do" (which is always "did" regardless of subject) to form the past simple tense. In the general case, this form is synonymous with the conjugated past simple ("lost"), but it typically has the connotation of either contradicting a prior negative (in this case "no I haven't"), and/or emphasizing that the event indeed happened. I can find no online reference defining this construction as anything other than past simple tense, and oddly I can't find any reference that uses the "do" verb to construct this tense in the affirmative (only the negative), but it definitely exists in modern speech.

The other possibility, which is less fitting, is the present perfect:

No, I haven't, but my brother has lost his camera on a trip once.

This is marginal; while it would work to express the fact that your brother, at one point in the indefinite past, lost a camera, the word "once" feels redundant; the use of the present perfect coupled with "once" makes it seem like the count of the number of times it's happened (once) is more important to the speaker than the event (losing a camera). If the sentence instead ended with "many times", this tense would be the only valid answer, but as it is it feels a little clunky.